Geometric convergence results for closed minimal surfaces via bubbling analysis

TL;DR

This paper establishes geometric convergence results for closed minimal surfaces using bubbling analysis, demonstrating compactness and genus behavior under curvature bounds and index constraints.

Contribution

It applies bubbling analysis to prove smooth convergence and genus drop phenomena for minimal surfaces with bounded index and genus or area, extending understanding of their geometric limits.

Findings

Sequential compactness of minimal surfaces of index one and fixed genus on positively curved 3-spheres.

Quantitative relation between genus drop and bubbling phenomena during convergence.

Sharp estimates on convergence multiplicity based on the number of bubble ends.

Abstract

We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus $\gamma$ is sequentially compact for any $\gamma\geq 1$. Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity $m\geq 1$, away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process.